Time headway in traffic flow is the elapsed time between the

“Time headway” in traffic flow is the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point. Let x = the time headway for two randomly chosen consecutive cars on a freeway during a period of heavy flow. The following probability density function (pdf) of x is

                  0                      otherwise

Find the value of A.

What is the probability that the time headway is at most 6 seconds?

Solution

a)

Here, the integral of f(x) over all x must be 1:

Integral [A exp(-.15(x-0.5)) dx] | (0.5,oo) = 1

(-1/0.15)*A exp(-.15(x-0.5)) | (0.5,oo) = 1

-(-1/0.15)*A = 1

A = 0.15 [ANSWER]

*****************

Now,

P(x<=6) = exp(-.15(x-0.5)) | (0.5,6)

= -exp(-.15(6-0.5)) - [-exp(-.15(0.5-0.5))]

= 0.561765008 [answer]